Joint distributions of the numbers of visits for finite-state Markov chains (Q1303853)

For a discrete-time Markov chain with finite state space \(\{1,\dots, r\}\) we consider the joint distribution of the numbers of visits in states \(1,\dots, r-1\) during the first \(N\) steps or before the \(N\)th visit to \(r\). From the explicit expressions for the corresponding generating functions we obtain the limiting multivariate distributions as \(N\to\infty\) when state \(r\) becomes asymptotically absorbing and for \(J= 1,\dots, r-1\) the probability of a transition from \(r\) to \(j\) is of order \(1/N\).